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Early Warning Signals of Environmental Tipping Points Early Warning Signals of Environmental Tipping Points

Early Warning Signals of Environmental Tipping Points - PowerPoint Presentation

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Early Warning Signals of Environmental Tipping Points - PPT Presentation

Chris Boulton 25 th April 2013 CABoultonexeteracuk cboulton89 cboulton89wordpresscom Tipping Points A bifurcation in the system where a stable equilibrium becomes unstable Consequently the system moves to a new stable state or regime ID: 262813

warning time system series time warning series system tipping point test early coefficient indicator estimation noise bifurcation fallacy alarms

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Slide1

Early Warning Signals of Environmental Tipping Points

Chris

Boulton

25

th

April 2013

C.A.Boulton@exeter.ac.uk

@cboulton_89

cboulton89.wordpress.com Slide2

Tipping Points

A bifurcation in the system where a stable equilibrium becomes unstable.

Consequently the system moves to a new stable state or regime.

Difficult to predict from observing time series without analysis.

There is usually a sense of irreversibility or hysteresis meaning more work would be needed to return to the previous stable state. Slide3

Causes

Forcing the system (climate) causes the approach to the bifurcation.

The system is also perturbed by noise (weather) which pushes the system out of the basin of attraction of the state.

In some cases, noise alone is enough to drive to system to a tipping point without approaching a bifurcation. EWS

should

not work in this case.Slide4

A Simple ExampleSlide5

ExampleSlide6

ExampleSlide7

ExampleSlide8

Generic Early Warning Signals

We look for a change in indicators over the time series.

A window length is specified to calculate the indicator in, which moves up one time point, creating a time series for the indicator.

Time series is usually

detrended

, especially if it drifts.

Kendall tau rank correlation coefficient is used to measure tendency of

indicator (1

if

always increases

and -1 if always decreasing).Slide9

AR(1) Coefficient Estimation

The dominant eigenvalue in the system tends to zero approaching a tipping point, causing critical slowing down.

This can be seen as an increase in the AR(1) coefficient ‘a’, where x(t+1) = a*x(t)+e (e - noise).

This can be seen as ‘today is becoming more like yesterday’.Slide10

Variance and Skewness

Variance ‘generally’ increases on the approach to the tipping point due to the destabilisation of the state.

In certain rare cases it can decrease (Dakos et al. 2012).

Skewness can increase or decrease depending on the direction the tipping point is going to move the system.

Dakos

et al. (2012)

Ecology

93(2) pp 264-271Slide11

Using a Generic IndicatorSlide12

Using AR(1) Coefficient EstimationSlide13

Using AR(1) Coefficient EstimationSlide14

Using AR(1) Coefficient EstimationSlide15

Robustness

In the animation, a window length (WL) of 400 (1/2 time series length) and a bandwidth (BW) of 30 was used during

detrending

.

Tau can be computed for combinations of WL and BW to check for robustness.

Decrease in variance at low bandwidths suggests ‘reddening’.Slide16

Hypothesis Testing

In the case where we have enough knowledge of the dynamics, we can create null models.

We can run an ensemble where the bifurcation is not approached and measure the

taus

on this.

Important to use same WL and BW.

p = 0.006Slide17

Hypothesis Testing

Most likely don’t know dynamics of the system and only have time series.

Can also test other null models derived from time series, such as bootstrapping, sampling from the same probability distribution and recreating the original time series using the statistics of the

detrended

time series (

Dakos

et al

, 2008).

Dakos

et al. (

2008)

PNAS

105, pp. 14308-14312Slide18

How Much Warning?

Especially in climate systems, it’s very important to know how much time there is for prevention. Maybe only adaptation is possible.

AR(1) coefficient estimation should reach 1 at tipping point but

detrending

and window length choices can change the value so we can only really look for increases.

Observing trends in hindsight and by confirming these with Kendall’s tau is simple.

What about in real time...?Slide19

How Much Warning?Slide20

How Much Warning?Slide21

How Much Warning?Slide22

How Much Warning?Slide23

How Much Warning?Slide24

How Much Warning?Slide25

False and Missed Alarms

As with nearly all forecasting, there is the chance of missed and false alarms.

In reality we don’t have unlimited time series to test early warning signals on.

To combat this, we try to test as many time series of a system as possible, test for robustness and test more than one indicator.Slide26

Prosecutor’s Fallacy

In law, a fallacy resulting from confusion that a suspect is guilty due to fitting evidence which is significant, i.e. The probability evidence is satisfied give the defendant is innocent, P(E|I) is small.

However, this doesn’t imply that the defendant is guilty given they match the evidence, P(I|E) is also small.

This can be proven by

Bayes

’ theorem.

A similar phenomenon occurs when looking for early warnings of tipping points.Slide27

Prosecutor’s Fallacy

Boettiger

& Hastings (2012) test a model which exhibits a bifurcation and creates an ensemble where this is not approached (no forcing).

Some pass the tipping point by chance (noise induced), the majority do not. Tau is calculated on all members.

Taus

from tipping members are in grey histogram.

Boettiger

& Hastings (2012)

Proc. R. Soc. BSlide28

Prosecutor’s Fallacy

We might be happy that our early warning works when we are approaching a tipping point.

However these are still false alarms that are caused by correlated noise pushing the system towards the tipping point. These systems tip quickly.

It could be argued that we are more likely to find an early warning signal in a time series which has tipped, because we have singled it out to test.Slide29

Conclusions

Early warnings of approaching bifurcations in systems can be determined due to critical slowing down observed in the time series of the system.

We can hypothesis test these the signals we find as well as test their robustness.

There is still the chance of false or missed alarms which we try to combat with the use of more than just one indicator.

We need to be wary of hindsight predictions and remember that observing early warning in real time could be difficult.